#151444

order	512
nilpotency class	5
isoclinism	isoclinic class of #151260
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2196
commutator, Frattini, cosocle	#391	C₂³
prime radical	C₂³	B(4,𝔽₂)
LCS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #391 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

264 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,8)(3,9,11,16)(4,12,14,7)(6,13,15,10)(17,18,23,29)(19,30,35,44)(20,36,32,24)(21,26,31,27)(22,28,40,47)(25,34,45,37)(33,42,48,43)(38,41,39,46);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19)(18,24)(20,31)(21,32)(22,34)(23,35)(25,41)(26,30)(27,44)(28,42)(29,36)(33,39)(37,40)(38,48)(43,47)(45,46);(17,20,22,33,23,32,40,48)(18,25,28,44,29,45,47,30)(19,31,34,39,35,21,37,38)(24,41,42,27,36,46,43,26)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,g^a=gh,d^b=df,e^b=eghi,h^b=hi,d^c=dg,e^c=ehi,h^c=hi,e^d=ehi,g^d=gi,g^e=gi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58640);